Parametric Curves and Surfaces
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) Text PARAMETRIC SURFACES
Parametric curves result from expressions for coordinates that depend on one variable. Suppose that these expressions depended instead on two variables, u and v. For a constant value of v, limiting variation to u alone, the resulting graph would be a parametric curve. For a slightly different value of v, the result would be another parametric curve similar to the first, but slightly moved or distorted. Considering all possible values of v, one arrives at a surface which is the set of all possible parametric curves for varying u and constant v.
One could consider parametric curves for varying v and constant u and arrive at the same result.
Demos
Parametric Surfaces
 
This demo portrays the principle discussed above—that one can understand parametric surfaces as twodimensional objects traced out by onedimensional parametric curves. The example used is known as a torus.
To see this torus (or any other surface you decide to parametrize) traced out, change a and b using the buttons [<<] and [>>].

Exercises Parametrize the following surfaces:
 The square in the xy plane bounded by the x and y axes, the line x = 1, and the line y = 1.
 The triangle in the xy plane bounded by the x and yaxes and the line x + y = 1.
 A cylinder (without its bases) of radius 2 and height 2 centered at the origin, with rotational symmetry about the zaxis.
 A sphere of radius 1 centered at the origin.
 A hemisphere of radius 1 centered at the origin, limited to nonnegative x values.
 A “monkey saddle” (z = x^3 – 3xy^2) where x and y are limited to the disk of radius 1 in the xy plane centered at the origin.
